RMS Speed of Gas Molecules Calculator

How it works

v_rms = √(3RT ÷ M) · v_avg = √(8RT ÷ πM) · v_p = √(2RT ÷ M) · R = 8.314462618 J/(mol·K), M in kg/mol · average kinetic energy per molecule = 3/2 k_B T, per mole = 3/2 RT

In the kinetic theory of gases, molecules move with a spread of speeds given by the Maxwell–Boltzmann distribution. The root-mean-square speed is the square root of the average of the squared speeds, so it ties directly to the average kinetic energy, 3/2 kT, which depends only on temperature. Lighter molecules therefore move faster at the same temperature: hydrogen is four times faster than oxygen. The three characteristic speeds always stand in the ratio v_p : v_avg : v_rms = 1 : 1.128 : 1.225.

Worked example

Nitrogen (M = 28.014 g/mol = 0.028014 kg/mol) at 25 °C (298.15 K): v_rms = √(3 × 8.314 × 298.15 ÷ 0.028014) = 515.2 m/s, about 1,855 km/h. Its mean speed is 474.7 m/s, the most probable speed 420.7 m/s, and each molecule carries on average 6.175 × 10⁻²¹ J of kinetic energy.

Sources: HyperPhysics — Kinetic theory: molecular speeds · LibreTexts Chemistry — Molecular speeds (Maxwell–Boltzmann)

Frequently asked questions